Periodic orbits in systems with backlash: Stability, classification & observability

Backlash, clearance or dead zone is a common feature of many mechanical systems and can undermine the performance of the system, since it has a large influence on the dynamics and control of systems. It can be caused by intended clearance necessary for assembly and operation, but may also be the result of operational wear and tear. Systems with backlash form a subclass of discontinuous mechanical systems and can be modeled as piecewise linear systems. In this work, both stiffness and damping are modeled with piecewise characteristics. A single and multiple degree-of-freedom model with backlash are analyzed for their harmonic periodic orbits as a function of excitation frequency and amplitude. The systems are modeled as tri-linear systems, with no stiffness in the backlash gap. This leads to a rigid body motion in this region. To calculate the flow of the piecewise linear systems, a simulation method is used that utilizes the knowledge of the analytical solutions for linear systems. This method also allows for analytical calculation of the fundamental solution matrix. This is beneficial for applying this simulation method in the multiple shooting method, which is used to calculate the periodic orbits. First, both the single and multiple degree-of-freedom system are characterized by their response diagram for a fixed excitation amplitude. Here, the amplitude of both stable and unstable periodic orbits are calculated. The response diagram shows a combination of branches that is characteristic for a hardening oscillator, with multiple solutions in some frequency ranges. The periodic orbits are characterized by their number of subspace boundary crossings in excitation frequency and amplitude plane. Next to the number of boundary crossings, the periodic orbits are characterized by the maximum absolute value of the Floquet multipliers. The Floquet multipliers jump when the number of subspace boundary crossings changes, so this characterization gives the same information. However, the classification by Floquet multipliers also distinguishes symmetric and asymmetric periodic orbits and therefore gives more information. These conclusions hold for both systems. When a system with backlash is used in practice, often only the output is measured. Information on the other states, especially the backlash gap, may however be relevant for analysis and control. Therefore, an observer is designed for the multiple degree-of-freedom system. Simulations of the observer show that it converges to an error much smaller as was expected. Yet, the convergence rate is low. Further research is needed to analyze the discrepancy between theory and simulations and to increase the performance of the observer

Cooperative look-ahead control for fuel-efficient and safe heavy-duty vehicle platooning

The operation of groups of heavy-duty vehicles (HDVs) at a small inter-vehicular distance (known as platoon) allows to lower the overall aerodynamic drag and, therefore, to reduce fuel consumption and greenhouse gas emissions. However, due to the large mass and limited engine power of HDVs, slopes have a significant impact on the feasible and optimal speed profiles that each vehicle can and should follow. Therefore maintaining a short inter-vehicular distance as required by platooning without coordination between vehicles can often result in inefficient or even unfeasible trajectories. In this paper we propose a two-layer control architecture for HDV platooning aimed to safely and fuel-efficiently coordinate the vehicles in the platoon. Here, the layers are responsible for the inclusion of preview information on road topography and the real-time control of the vehicles, respectively. Within this architecture, dynamic programming is used to compute the fuel-optimal speed profile for the entire platoon and a distributed model predictive control framework is developed for the real-time control of the vehicles. The effectiveness of the proposed controller is analyzed by means of simulations of several realistic scenarios that suggest a possible fuel saving of up to 12% for the follower vehicles compared to the use of standard platoon controllers.Comment: 16 pages, 16 figures, submitted to journa

Model reduction of networked passive systems through clustering

In this paper, a model reduction procedure for a network of interconnected identical passive subsystems is presented. Here, rather than performing model reduction on the subsystems, adjacent subsystems are clustered, leading to a reduced-order networked system that allows for a convenient physical interpretation. The identification of the subsystems to be clustered is performed through controllability and observability analysis of an associated edge system and it is shown that the property of synchronization (i.e., the convergence of trajectories of the subsystems to each other) is preserved during reduction. The results are illustrated by means of an example.Comment: 7 pages, 2 figures; minor changes in the final version, as accepted for publication at the 13th European Control Conference, Strasbourg, Franc

Balanced Truncation of Networked Linear Passive Systems

This paper studies model order reduction of multi-agent systems consisting of identical linear passive subsystems, where the interconnection topology is characterized by an undirected weighted graph. Balanced truncation based on a pair of specifically selected generalized Gramians is implemented on the asymptotically stable part of the full-order network model, which leads to a reduced-order system preserving the passivity of each subsystem. Moreover, it is proven that there exists a coordinate transformation to convert the resulting reduced-order model to a state-space model of Laplacian dynamics. Thus, the proposed method simultaneously reduces the complexity of the network structure and individual agent dynamics, and it preserves the passivity of the subsystems and the synchronization of the network. Moreover, it allows for the a priori computation of a bound on the approximation error. Finally, the feasibility of the method is demonstrated by an example

Path-based Stability Analysis for Monotone Control Systems on Proper Cones

In this paper, we study positive invariance and attractivity properties for nonlinear control systems which are monotone with respect to proper cones. Monotonicity simplifies such analysis for specific sets defined by the proper cones. Instead of Lyapunov functions, a pair of so called paths in the state space and input space play important roles. As applications, our results are utilized for analysis of asymptotic stability and also input-to-state stability on proper cones. The results are illustrated by means of examples

Scalable robustness of interconnected systems subject to structural changes

This paper studies the robustness of large-scale interconnected systems with respect to external disturbances, focussing on their scalability properties. Specifically, a notion of scalability is introduced that asks for these robustness properties to remain unchanged under a structural change of the system, such as the addition/removal of a subsystem or a change in the interconnection structure. Both necessary and sufficient conditions, in terms of the interconnection structure and edge weights, are given under which elementary structural changes are scalable. The results are illustrated through a simple example.Comment: Published in the Proceedings of the IFAC World Congress, Berlin, Germany, 202

Contraction Analysis of Monotone Systems via Separable Functions

In this paper, we study incremental stability of monotone nonlinear systems through contraction analysis. We provide sufficient conditions for incremental asymptotic stability in terms of the Lie derivatives of differential one-forms or Lie brackets of vector fields. These conditions can be viewed as sum- or max-separable conditions, respectively. For incremental exponential stability, we show that the existence of such separable functions is both necessary and sufficient under standard assumptions for the converse Lyapunov theorem of exponential stability. As a by-product, we also provide necessary and sufficient conditions for exponential stability of positive linear time-varying systems. The results are illustrated through examples